Universal finite-size scaling in the extraordinary-log boundary phase of three-dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mat

Recent advances in boundary critical phenomena have led to the discovery of a new surface universality class in the three-dimensional <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>O</a:mi><a:mo>(</a:mo><a:mi>N</a:mi><a:mo>)</a:mo></a:mrow></a:math> model. The newly found “extraordinary-log” phase can be realized on a two-dimensional surface for <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>N</b:mi><b:mo>&lt;</b:mo><b:msub><b:mi>N</b:mi><b:mi>c</b:mi></b:msub></b:mrow></b:math>, with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:msub><c:mi>N</c:mi><c:mi>c</c:mi></c:msub><c:mo>&gt;</c:mo><c:mn>3</c:mn></c:mrow></c:math>, and on a plane defect embedded into a three-dimensional system, for any <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>N</d:mi></d:math>. One of the key features of the extraordinary-log phase is the presence of logarithmic violations of standard finite-size scaling. In this work we study finite-size scaling in the extraordinary-log universality class by means of Monte Carlo simulations of an improved lattice model. We simulate the model with open boundary conditions, realizing the extraordinary-log phase on the surface for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>N</e:mi><e:mo>=</e:mo><e:mn>2</e:mn><e:mo>,</e:mo><e:mn>3</e:mn></e:mrow></e:math>, as well as with fully periodic boundary conditions and in the presence of a plane defect for <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>N</f:mi><f:mo>=</f:mo><f:mn>2</f:mn><f:mo>,</f:mo><f:mn>3</f:mn><f:mo>,</f:mo><f:mn>4</f:mn></f:mrow></f:math>. In line with theory predictions, renormalization-group invariant observables studied here exhibit a logarithmic dependence on the size of the system. We numerically access not only the leading term in the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>β</g:mi></g:math> function governing these logarithmic violations, but also the subleading term, which controls the evolution of the boundary phase diagram as a function of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>N</h:mi></h:math>.